7a^2-18a+5=15

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Solution for 7a^2-18a+5=15 equation:


Simplifying
7a2 + -18a + 5 = 15

Reorder the terms:
5 + -18a + 7a2 = 15

Solving
5 + -18a + 7a2 = 15

Solving for variable 'a'.

Reorder the terms:
5 + -15 + -18a + 7a2 = 15 + -15

Combine like terms: 5 + -15 = -10
-10 + -18a + 7a2 = 15 + -15

Combine like terms: 15 + -15 = 0
-10 + -18a + 7a2 = 0

Begin completing the square.  Divide all terms by
7 the coefficient of the squared term: 

Divide each side by '7'.
-1.428571429 + -2.571428571a + a2 = 0

Move the constant term to the right:

Add '1.428571429' to each side of the equation.
-1.428571429 + -2.571428571a + 1.428571429 + a2 = 0 + 1.428571429

Reorder the terms:
-1.428571429 + 1.428571429 + -2.571428571a + a2 = 0 + 1.428571429

Combine like terms: -1.428571429 + 1.428571429 = 0.000000000
0.000000000 + -2.571428571a + a2 = 0 + 1.428571429
-2.571428571a + a2 = 0 + 1.428571429

Combine like terms: 0 + 1.428571429 = 1.428571429
-2.571428571a + a2 = 1.428571429

The a term is -2.571428571a.  Take half its coefficient (-1.285714286).
Square it (1.653061225) and add it to both sides.

Add '1.653061225' to each side of the equation.
-2.571428571a + 1.653061225 + a2 = 1.428571429 + 1.653061225

Reorder the terms:
1.653061225 + -2.571428571a + a2 = 1.428571429 + 1.653061225

Combine like terms: 1.428571429 + 1.653061225 = 3.081632654
1.653061225 + -2.571428571a + a2 = 3.081632654

Factor a perfect square on the left side:
(a + -1.285714286)(a + -1.285714286) = 3.081632654

Calculate the square root of the right side: 1.755457961

Break this problem into two subproblems by setting 
(a + -1.285714286) equal to 1.755457961 and -1.755457961.

Subproblem 1

a + -1.285714286 = 1.755457961 Simplifying a + -1.285714286 = 1.755457961 Reorder the terms: -1.285714286 + a = 1.755457961 Solving -1.285714286 + a = 1.755457961 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.285714286' to each side of the equation. -1.285714286 + 1.285714286 + a = 1.755457961 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + a = 1.755457961 + 1.285714286 a = 1.755457961 + 1.285714286 Combine like terms: 1.755457961 + 1.285714286 = 3.041172247 a = 3.041172247 Simplifying a = 3.041172247

Subproblem 2

a + -1.285714286 = -1.755457961 Simplifying a + -1.285714286 = -1.755457961 Reorder the terms: -1.285714286 + a = -1.755457961 Solving -1.285714286 + a = -1.755457961 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.285714286' to each side of the equation. -1.285714286 + 1.285714286 + a = -1.755457961 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + a = -1.755457961 + 1.285714286 a = -1.755457961 + 1.285714286 Combine like terms: -1.755457961 + 1.285714286 = -0.469743675 a = -0.469743675 Simplifying a = -0.469743675

Solution

The solution to the problem is based on the solutions from the subproblems. a = {3.041172247, -0.469743675}

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